I keep finding the standard overhand knot comes undone as I rap down, have other people found this?
I've started using a double or triple (shot below, I always get nervous on the new mirrorball rap) overhand/barrel knot.

Sometimes I tie both ropes together, when it's a clean toss down a steep wall. If it's a slabby blocky abseil I tie and toss them them seperately.

I started going for full overkill with a figure 8 on a bight, it's less likely to magically pull through the device, and if you do reach the end of the rope you can clip in or stand in the loops. Also when you're pulling the rope and rethreading you can clip it to your harness, in case you're feeling particularly rope-droppingly retarded on the day.

I've been using the stopper knot you've shown in the picture, it should be solid, I think it's just half of a double or tripple fishermans knot. An overhand seems a little insecure as you've found, might only be good for a false sense of security.

Yes, I've found figure of eights come undone when they flap around, the double or triple stopper works for me.
Mostly I don't tie the ends together these days. Used to, but when it goes wrong it's a real mess.

If I'm using a stopper knot I use a figure-8 on the bight. It hasn't seemed to untie in my experience and is really simple and way bulky - also gives you a loop to try and hook if you forget to untie the knot before starting to pull the rope.